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230-611-lm-4904 


BULLETIN 


OF 


THE  UNIVERSITY  OF  TEXAS 


Number  193 
Pour  Times  a  Month. 


OFFICIAL  SERIES  NO.  58 


AUGUST  1,  1911 


Mathematics  In  the  High  School 


PUBLISHED  BY 

THE  UNIVERSITY  OP  TEXAS 

AUSTIN,   TEXAS 

Entered  as  second-class  mail  matter  at  the  postoffice  at 
Austin,  Texas 


OUTLINE  OF  COURSE  OF  STUDY  IN  MATHEMATICS 

The  preparation  of  this  Bulletin  has  been  undertaken  with  a 
view  to  suggesting  the  most  economical  arrangement  of  the 
course  of  study  in  the  high  schools  in  order  that  students,  whether 
they  expect  to  continue  their  stupes  elsewhere  or  have  no  such 
intention,  may  get  the  greatest  benefit. 

The  bulletin  deals  with  methods  of  instruction  in  algebra 
and  will  be  followed  by  a  bulletin  on  the  teaching  of  geometry 
and  one  on  trigonometry.  A  synoptic  arrangement  of  the  course 
proposed  will  be  found  on  the  last  pages. 

Beginning  Algebra.  The  transition  from  arithmetic  to  algebra 
should  be  gradual.  Letters  should  be  introduced  to  represent 
numbers  during  the  last  half  year's  work  in  arithmetic  where- 
ever  such  introduction  is  advantageous.  The  formulas  of  per- 
centage and  discount  thus  become  more  compact  and  the  solu- 
tion of  many  problems  can  be  thus  simplified  by  the  use  of  equa- 
tions. Practically  all  arithmetics  do  use  letters  in  the  formula 
for  interest,  etc.,  but  no  algebraic  calculations  are  made  with 
these  letters  and  they  are  used  as  mere  aids  to  memory.  The 
pupil  is  expected  to  replace  them  at  once  with  numbers.  On  the 
contrary  a  beginning  is  made  in  algebra  when  we  regard  the 
formula  A=p  (1-j-rt) 

as  an  equation  when  either  p,  r  or  t  is  unknown  and  is  to  be 
found  in  terms  of  the  remaining  letters.  In  practical  problems 
of  this  sort  the  unknown  will  always  turn  out  to  be  a  positive 
number  and  at  this  stage  of  the  work  only  such  numbers  should 
be  introduced.  Finally  just  before  the  pupil  takes  up  the  actual 
text  in  algebra  simple  equations  involving  negative  solutions 
which  arise  naturally  should  be  taken  up  and  the  negative  re- 
sults interpreted  in  terms  of  the  student's  practical  experience. 
Thus  the  transition  from  the  ordinary  natural  numbers,  integral 
and  fractional,  which  are  either  positive  or  negative  to  the 
algebraic  numbers,  will  seem  less  startling  and  the  pupil  can  be 
led  to  realize  that  he  is  dealing  with  numbers  which  are  only 
generalizations  of  those  with  which  he  is  familiar. 

327838 


4  University  of  Texas  Bulletin 

FIRST  YEAR  IN  ALGEBRA 

Outline.    ,The  following  is  suggested: 

(1)  After  the  preliminary  training  in  literal  arithmetic  and 
the  simple  notions  of  the  equation  have  been  given,  the  actual 
work  of  the  first  year  should  be  begun.    At  this  place  zero  and 
the  negative  numbers  should  be  introduced.     The  origin  and 
necessity  of  these  should  be  shown.    The  discussion  should  begin 
with  natural  numbers  and  the  relative  magnitude  of  negative 
numbers  shown.    Use  concrete  illustrations  for  the  number  sys- 
tem, including  the  positive,  negative  and  zero  numbers.    At  this 
place  discuss  the  four  fundamental  operations  with  negative 
numbers. 

(2)  The  four  fundamental  operations  and  the  use  of  symbols. 
This  should  consist  of  operations  with  simple  polynomials,  the 
difficult  or  long  problems  under  this  heading  should  be  omitted 
until  after  the  simultaneous  solutions  of  simple  equations.    Al- 
ternate problems  with  literal  notation  with  problems  with  Arabic 
numbers  only.    For  example: 

3+4 
4—2 


12+16 
—6—8 

12+10—8  7X2=14 

and  show  how  by  means  of  this  problem  the  products  of 

(a+b)  (c— d) 

may  be  verified.  In  the  beginning  of  this  work  give  practice  in 
the  substitution  of  numbers  for  letters  in  literal  expressions  and 
identities. 

(3)  A  knowledge  of  simple  fractions  with  numerical  denom- 
inators.    The  elementary  processes  with  these  fractions  should 
be  given  before  or  together  with  the  solution  of  simple  equations. 

(4)  Solution  of  simple  equations  having  one  unknown.    The 
statement  of  problems.     Numerous  problems  should  be  solved 
with  explanation  by  the  pupil.     The  written  work  without  the 
explanation  of  each  step  is  not  sufficient.     The  solution  should 
always  be  given  in  such  a  manner  that  each  step  will  fully 
explain  itself.    For  example,  given 


Mathematics  in  the  High  School 


Subtracting  18  from  both  sides  we  have 

3x=39 
Dividing  both  sides  by  3 

x=13. 

As  soon  as  the  solution  of  equations  is  fully  understood,  simple 
problems  should  be  given.  The  statements  should  be  clearly 
given  in  details.  Let  it  be  sho^vm  in  every  statement  that  the 
letter  representing  the  unknown  is  a  number. 

(5)  Solution  of  equations  with  two  and  three  unknowns. 
In  this  work  the  equations  given  and  derived  should  be  num- 
bered and  details  of  the  solution  noticed  at  each  step.    The  usual 
three  methods  of  solution  should  be  equally  stressed.     Practice 
in  substitution  should  be  constantly  given.     If  one  unknown 
remains  to  be  found  use  the  method  of  substitution. 

(6)  ,The  graph  of  types  having  the  form  y=ax+b  where  a 
and  &  are  numbers.     The  solution  of  simultaneous  equations  of 
two  unknowns  to  be  illustrated  by  means  of  the  intersection  of 
two  loci. 

Give  simple  functions,  and  illustrate  by  means  of  the  graph. 

(7)  Longer  problems  in  the   four  fundamental  operations 
with  polynomials. 

(8)  Factors  of  simple  types,  especially  those  of  the  second 
degree.     In  learning  to  write  out  factors,  the  pupil  is  urged  to 
check  his  results  by  multiplication. 

(9)  Common  divisors  and  common  multiples  by  the  use  of 
factors.    In  order  that  these  two  processes  may  'be  differentiated 
from  each  other,  it  is  well  to  find  the  H.  C.  F.  and  L.  C.  M.  at 
the  same  time  of  any  given  set  of  numbers. 

(10)  The  square  root  of  simple  polynomials  by  the  use  of 
the  perfect  square    (a-j-b)2=a2-f-2ab+b2,  and  then  give  con- 
siderable practice  in  the  square  root  of  numbers. 

(11)  Solutions  of  quadratics  of  one  unknown.    Let  the  pupil 
see  the  reason  or  logic  in  the  solution.     Have  pupil  to  complete 
the  square  in  the  first  year's  work  rather  than  solve  by  the  so 
called  formula  method.     Stress  the  solution  by  means  of  factor- 
ing.   As  a  matter  of  practice  solve  by  both  methods  and  develop 
on  the  part  of  the  pupil  the  power  to  detect  the  easier  method 
for  any  given  problem. 


6  University  of  Texas  Bulletin 

(12)  Graphical  representation  of  the  type  y=a-|-bx-}-cx2. 
The  intersection  of  graphs  to  be  used  to  show  the  simultaneous 
solution.    Give  difinition  of  function  and  illustrate  by  means  of 
the  graph.     Introduce  the  functional  symbols. 

(13)  Practice  in  the  use  of  positive,  fractional,  negative  and 
zero  exponents  giving  as  much  of  the  theory  of  indices  as  prac- 
ticable in  the  first  year.    The  laws  of  integral  exponents  should 
be  clearly  given. 

(14)  Surds  and  Radicals  should  be  given  as  particular  ex- 
pressions for  exponents. 

METHODS 

Identities,  checks.  Let  us  now  suppose  that  the  pupil  is 
familiar  with  the  representation  of  numbers  by  letters  and  the 
rules  for  addition,  subtraction  and  multiplication.  He  is  in  a 
position  to  acquire  some  facility  and  accuracy  in  calculating 
with  algebraic  expressions.  He  should  be  taught  to  factor  simple 
expressions,  form  the  squares  of  binomials,  find  H.  C.  F.  and 
L.  C.  M.  by  factor  methods  and  handle  fractions. 

All  such  work  should  be  checked  by  the  direct  processes.  Later 
when  indices  are  taken  up  accuracy  should  be  checked  by  sub- 
stituting numbers  for  the  letters  involved.  Here  much  is  gained 
by  replacing  all  radical  signs  by  fractional  exponents  and  citing 
at  each  reduction  the  rule  for  indices  which  has  been  employed. 
In  this  way  such  erroneous  inductions  as  (a+b)$=ai+bi  can 
be  easily  eradicated.  .The  pupil  should  never  lose  sight  of  the 
fact  that  the  letters  represent  numbers  and  that  equality  means 
identity,  i.  e.  true  for  all  values  of  the  letters  involved  if  the 

2          L  2 

operations  have  a  meaning.     Thus  -  — — =a-|-b  for  all  values 

a — b 
of  a  and  b  except  a=b. 

The  fact  that  the  only  thing  that  can  be  done  to  a  fraction 
that  does  not  change  its  value  is  to  multiply  numerator  and 
denominator  by  the  same  factor  should  be  insisted  on. 

Treatment  of  the  Equation.  The  distinction  between  the 
equation,  say  3x — 5=0,  and  the  identity  x2 — 2x+l=(x — I)2 
should  be  pointed  out  and  by  substituting  various  values  of  x  in 
each,  it  should  be  shown  that  the  second  is  always  true  while 
the  first  is  not.  Roots  of  an  equation  should  be  defined  as  those 


Mathematics  in  the  High  School  7- 

values  of  the  unknown  that  satisfy  it.  From  the  fact  that  a 
product  is  zero  when  and  only  when  at  least  one  of  its  factors  is 
zero,  the  roots  of  such  expressions  as  3x(x  —  2)  (x4-7)=0  should 
be  found.. 

It  not  infrequently  happens  that  pupils  have  so  far  forgotten 
the  very  meanings  of  the  terms  they  use  that  when  asked  to 
solve,  say  3(x  —  1)  (x-j-2)=0  they  will  multiply  out,  complete 
square  and  solve  as  a  quadratic^  of  ten  arriving  at  an  incorrect 
solution!  .This  is  merely  the  result  of  unintelligent  formalism 
and  unthinking  routine. 

The  treatment  of  the  equation  is  made  the  central  feature  of 
most  texts  in  algebra,  to  such  an  extent  that  an  algebraic  ex- 
pression which  is  not  equated  to  zero  or  which  does  not  involve 
an  equality  sign  is  meaningless  to  many  pupils.  Such  pupils  if 
asked  to  add  several  fractions  do  so  by  first  "clearing  of  frac- 
tions," adding  the  numerators  and  having  thrown  away  the 
common  denominator  proudly  exhibit  a  polynomial  as  the  result 
of  their  labors  ! 

Board  work.  In  putting  work  on  the  board  for  students  terms 
should  not  be  "transposed"  from  one  side  of  the  equation  to 
the  other  by  changing  sign,  but  the  step  should  be  performed  by 
adding  the  same  expression  to  both  sides.  Instead  of  "clearing" 
of  fractions  it  is  better  to  reduce  all  the  fractions  to  a  common 
denominator  and  then  make  use  of  the  fact  that  a  fraction  is 
zero  when  its  numerator  is  zero,  provided  the  denominator  is 
not  zero  at  the  same  time.  All  calculations  should  be  made  by 
passing  from  one  step  to  the  next  by  means  of  an  identity 
familiar  to  the  pupil. 

The  solution  of  quadratics  in  the  second  year  is  best  taught 
by  the  factor  method,  e.  g. 

ax*  +bx+c=a(  x2  +x+   )=  a 


Thus  the  quadratic  can  be  zero  when  and  only  when 


=     or 


The  question  of  equivalence  of  equations  is  the  most  difficult 
that  the  teacher  of  algebra  has  to  contend  with.  When  dis- 
cussions of  this  sort  are  unavoidable  direct  substitution  back  in 


8  University  of  Texas  Bulletin 

the  original  equations  can  be  resorted  to  if  no  better  method 
suggests  itself.  Problems  of  this  sort  offer  the  best  opportunities 
for  independent  thought  and  should  be  given  careful  attention. 

Graphical  Illustrations.  Graphical  illustrations  are  now  freely 
used  in  all  modern  texts  to  illustrate  common  solutions  in  the 
case  of  simultaneous  equations,  double,  imaginary,  and  unequal 
roots  of  quadratics  and  the  point  plotting  of  curves  forms  a  grow- 
ing feature  of  such  texts.  The  graph  of  such  functions  as 

y=ax+b  y=ax2+bx-}-c  (a,  b,  and  c  numerical) 
should  be  carefully  studied  and  the  graphs  used  for  example  to 
show  if  b2 — 4ac  is  negative  the  polynomial  ax2-(-bx-)-c  always 
has  the  same  sign  as  a.  Graphs  serve  another  most  useful  pur- 
pose in  that  they  stress  the  numerical  significance  of  algebraic 
expressions,  feature  such  ideas  as  "excluded"  values  of  x  and  y 
and  make  good  drill  in  computation.  Point  plotting  can  degene- 
rate into  waste  of  time  however  unless  it  has  some  definite  end  in 
view  and  is  worse  than  waste  of  time  if  the  examples  given  re- 
quire more  knowledge  that  the  student  can  be  expected,  to  possess. 

Limits.  The  space  given  to  this  important  topic  is  necessarily 
small  in  elementary  texts,  it  is  however  a  notion  not  difficult  to 
acquire  and  involves  no  obscurity  especially  if  the  arithmetic 
sense  has  been  properly  developed. 

Graphical  methods  can  be  profitably  employed  in  this  con- 
nection, thus  the  student  could  plot  such  functions  as 

x 


x' 

(The  limits  approached  by  y  when  x  increases  indefinitely 
would  give  a  good  idea  of  the  limit  of  y  when  x  varies  as  indi- 
cated. 

Owing  to  the  loose  and  misleading  notation  of  many  texts 
students  have  generally  very  erroneous  ideas  concerning  zero. 
There  is  but  one  thing  to  do  here  and  that  is  to  make  the  em- 
phatic assertion  that  a  product  is  zero  when  and  only  when  one 
of  its  factors  is  zero  and  that  division  by  zero  is  always  impos- 
sible. The  behavior  of  the  function  y— ^  when  x  approaches 


Mathematics  in  the  High  School  9 

zero  as  limit  could  be  illustrated  by  a  graph  and  the  student 
told  that  for  x=0  there  is  no  value  of  y  given  by  the  expression  1 

X* 

Supplementary  Topics.  Inequalities,  variation,  ratio  and  pro- 
portion, arithmetic  and  geometric  progressions,  expansion  of 
(a+b)n  for  n=l,  2,  3,  4,  5  found  by  direct  multiplication,  and 
so  much  of  the  theory  of  logarithms  as  follows  immediately  from 
the  index  laws  should  follow  the  work  just  outlined. 

Excluded  Topics.  Such  topics  «s  H.  C.  F.  by  Euclid's  method 
of  continued  division,  decomposition  into  partial  fractions,  har- 
monic means  and  progressions,  indeterminate  coefficients  (so 
called),  proofs  of  the  binomial  theorem  for  any  positive  integral 
power,  permutations  and  combinations  should  be  omitted  in  a 
course  which  occupies  only  a  year  and  a  half.  Attention  being 
concentrated  on  the  most  fundamental  processes. 

A  SECOND  COURSE  IN  ALGEBRA 

This  should  follow  the  year's  work  in  Plane  Geometry. 

(1)  Rapid  review  of  first  four  fundamental  operations. 

(2)  Simultaneous  solutions,  graphical  representation  of  the 
same.     Simple  use  of  determinants. 

(3)  Extraction  of  roots  square  and  cube. 

(4)  Solution  of  quadratics.    Graphics  of  the  same. 

(5)  Theory  of  exponents.     Surds  to  be  treated  as  cases  of 
fractional  exponents.    Imaginaries. 

(6)  Theory  of  quadratics. 

(7)  Variation,  ratio  and  proportion. 

(8)  Arithmetical  and  geometrical  progressions. 

(9)  Binomial  theorems  for  integral  exponents. 

(10)  Logarithms. 

(11)  Limits. 

COURSE  OF  STUDY 

The  order  or  arangement  of  mathematical  subjects  in  the  high 
school  course  suggested  here  is  found  in  the  strongest  and  best 
high  schools  of  the  Middle  West,  and  is  used  in  the  main  by  the 
high  schools  of  the  East.  It  gives  a  more  rational  development 
and  training  in  mathematics  than  is  found  in  many  of  our  courses 
of  study  found  in  Texas  schools.  It  is  given  below,  as  follows : 

(1)     One  year's  work  in  Elementary  Algebra. 


10  University  of  Texas  Bulletin 

This  should  be  preceded  by  a  preliminary  course  in  Literal 
Arithmetic,  as  suggested  above. 

(2)  The  year's  work  in  Algebra  should  be  followed  by  a 
year  in  Plane  Geometry. 

In  this  year  of  Geometry  Algebraic  applications  should  be  con- 
stantly given. 

(3)  Following  the  Plane  Geometry  at  least  a  half  year  in 
Algebra  should  be  given.  This  should  consist  of  a  review  of  the 
first  year 's  work  with  a  few  advanced  topics  in  addition. 

(4)  A  half  year  in  Solid  Geometry. 

Much  work  should  be  given  in  the  solution  and  computation  of 
problems  by  the  use  of  Algebra. 

(5)  A  half  year  in  Plane  Trigonometry. 

M.  B.  PORTER, 
Professor  of  Pure  Mathematics. 

C.  0.  RICE, 

Adjunct   Professor    of   Applied 
Mathematics. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN     INITIAL    FINE     OF     25     CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  S1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


NOV      2    1932 


LD  21-50m-8,'32 


327838 


5696* 


y(J-t 


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